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 Incoming links: Joint distribution function

Joint probability density Created 2026-09-29 Updated 2026-10-05

f_{X, Y}

satisfies

P ((X, Y) \in A) = \iint_{A} f_{X, Y} (x, y) d x d y .

Its coordinate integrals are the marginal densities, and integrating it over a rectangle gives the corresponding joint distribution function.

Write

Φ

B_{t} \leq M_{t}

, for

y \geq 0

and

x > y

P (B_{t} \leq x, M_{t} \leq y) = P (M_{t} \leq y) = 2Φ (\frac{y}{t}) - 1.

For

x \leq y

P (B_{t} \leq x, M_{t} \leq y) = P (B_{t} \leq x) - P (B_{t} \leq x, M_{t} > y) = Φ (\frac{x}{t}) - P (B_{t} > 2 y - x) = Φ (\frac{x}{t}) + Φ (\frac{2 y - x}{t}) - 1.

F_{B_{t}, M_{t}} (x, y) = ⎩ ⎨ ⎧ 0, Φ (x / t) + Φ ((2 y - x) / t) - 1, 2Φ (y / t) - 1, y < 0, y \geq 0, x \leq y, y \geq 0, x > y .